Sadeghimanesh, A. | Search bioRxiv

bioRxiv2021

Unidirectional migration of populations with Allee effect

In this note we consider two populations living on identical patches, connected by unidirectional migration, and subject to strong Allee effect. We show that by increasing the migration rate, there are more bifurcation sequences than previous works showed. In particular, the number of steady states can change from 9 (small migration) to 3 (large migration) at a single bifurcation point, or via a sequences of bifurcations with the system having 9,7,5,3 or 9,7,9,3 steady states, depending on the Allee threshold. This is in contrast with the case of bidirectional migration, where the number of steady states always goes through the same bifurcation sequence of 9,5,3 steady states as we increase the migration rate, regardless of the value of the Allee threshold.

ecology↗

bioRxiv2021

Exotic bifurcations in three connected populations with Allee effect

We consider three connected populations with strong Allee effect, and give a complete classification of the steady state structure of the system with respect to the Allee threshold and the dispersal rate, describing the bifurcations at each critical point where the number of steady states change. One may expect that by increasing the dispersal rate between the patches, the system would become more well-mixed hence simpler. However, we show that it is not always the case, and the number of steady states may (temporarily) increase by increasing the dispersal rate. Besides sequences of pitchfork and saddle-node bifurcations, we find triple-transcritical bifurcations and also a sun-ray shaped bifurcation where twelve steady states meet at a single point then disappear. The major tool of our investigations is a novel algorithm that decomposes the parameter space with respect to the number of steady states and find the bifurcation values using cylindrical algebraic decomposition with respect to the discriminant variety of the polynomial system.

ecology↗